@Article{ATA-36-326,
author = {Jiu , Jiaxi and Li , Zhongkai},
title = {Boundary Values of Generalized Harmonic Functions Associated with the Rank-One Dunkl Operator},
journal = {Analysis in Theory and Applications},
year = {2020},
volume = {36},
number = {3},
pages = {326--347},
abstract = {<p style="text-align: justify;">We consider the local boundary values of generalized harmonic functions associated with the rank-one Dunkl operator $D$ in the upper half-plane $R^{2}_+=R\times(0,\infty)$, where<br/>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;$$(Df)(x)=f&#39;(x)+(\lambda/x)[f(x)-f(-x)]$$ <br/>for given $\lambda\ge0$. A $C^2$ function $u$ in $R^{2}_+$ is said to be $\lambda$-harmonic if $(D_x^2+\partial_{y}^2)u=0$. For a $\lambda$-harmonic function $u$ in $R^{2}_+$ and for a subset $E$ of $\partial R^{2}_+=R$ symmetric about $y$-axis, we prove that the following three assertions are equivalent: (i) $u$ has a finite non-tangential limit at $(x,0)$ for a.e. $x\in E$; (ii) $u$ is non-tangentially bounded for a.e. $x\in E$; (iii) $(Su)(x)&lt;\infty$ for a.e. $x\in E$, where $S$ is a Lusin-type area integral associated with the Dunkl operator $D$.</p>},
issn = {1573-8175},
doi = {https://doi.org/10.4208/ata.OA-SU11},
url = {http://global-sci.org/intro/article_detail/ata/18289.html}
}